Galois Representations, $ L$-series and a Conjecture of Artin

The Galois group $ \Gal (\overline{\mathbf{Q}}/\mathbf{Q})$ is an object of central importance in number theory, and we can interpreted much of number theory as the study of this group. A good way to study a group is to study how it acts on various objects, that is, to study its representations.

Endow $ {\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$ with the topology which has as a basis of open neighborhoods of the origin the subgroups $ \Gal (\overline{\mathbf{Q}}/K)$, where $ K$ varies over finite Galois extensions of  $ \mathbf {Q}$. (Note: This is not the topology got by taking as a basis of open neighborhoods the collection of finite-index normal subgroups of $ {\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$.) Fix a positive integer $ n$ and let $ \GL _n(\mathbf{C})$ be the group of $ n\times n$ invertible matrices over  $ \mathbf{C}$ with the discrete topology.

Definition 9.5.1   A complex $ n$-dimensional representation of $ {\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})$ is a continuous homomorphism

$\displaystyle \rho:{\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to \GL _n(\mathbf{C}).

For $ \rho$ to be continuous means that if $ K$ is the fixed field of $ \Ker (\rho)$, then $ K/\mathbf{Q}$ is a finite Galois extension. We have a diagram

$\displaystyle \xymatrix{ {{\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})}\ar[...
...[dr]& &{\GL _n(\mathbf{C})}\\
&{\Gal (K/\mathbf{Q})}\ar@{^{(}->}[ur]_{\rho'}}

Remark 9.5.2   That $ \rho$ is continuous implies that the image of $ \rho$ is finite, but the converse is not true. Using Zorn's lemma, one can show that there are homomorphisms $ {\mathrm{Gal}}(\overline{\mathbf{Q}}/\mathbf{Q})\to\{\pm 1\}$ with image of order $ 2$ that are not continuous, since they do not factor through the Galois group of any finite Galois extension.

Fix a Galois representation $ \rho$ and let $ K$ be the fixed field of $ \ker(\rho)$, so $ \rho$ factors through $ \Gal (K/\mathbf{Q})$. For each prime $ p\in\mathbf{Z}$ that is not ramified in $ K$, there is an element $ \Frob _\mathfrak{p}\in\Gal (K/\mathbf{Q})$ that is well-defined up to conjugation by elements of $ \Gal (K/\mathbf{Q})$. This means that $ \rho'(\Frob _p)\in
\GL _n(\mathbf{C})$ is well-defined up to conjugation. Thus the characteristic polynomial $ F_p(x)\in\mathbf{C}[x]$ of $ \rho'(\Frob _p)$ is a well-defined invariant of $ p$ and $ \rho$. Let

$\displaystyle R_p(x) = x^{\deg(F_p)}\cdot F_p(1/x) = 1 + \cdots +
\det(\Frob _p)\cdot x^{\deg(F_p)}$

be the polynomial obtain by reversing the order of the coefficients of $ F_p$. Following E. Artin [Art23,Art30], set

$\displaystyle L(\rho,s) = \prod_{p\text{ unramified}} \frac{1}{R_p(p^{-s})}.$ (9.3)

We view $ L(\rho,s)$ as a function of a single complex variable $ s$. One can prove that $ L(\rho,s)$ is holomorphic on some right half plane, and extends to a meromorphic function on all $ \mathbf{C}$.

Conjecture 9.5.3 (Artin)   The $ L$-function of any continuous representation

$\displaystyle \Gal (\overline{\mathbf{Q}}/\mathbf{Q})\to\GL _n(\mathbf{C})$

is an entire function on all $ \mathbf{C}$, except possibly at $ 1$.

This conjecture asserts that there is some way to analytically continue $ L(\rho,s)$ to the whole complex plane, except possibly at $ 1$. (A standard fact from complex analysis is that this analytic continuation must be unique.) The simple pole at $ s=1$ corresponds to the trivial representation (the Riemann zeta function), and if $ n\geq 2$ and $ \rho$ is irreducible, then the conjecture is that $ \rho$ extends to a holomorphic function on all $ \mathbf{C}$.

The conjecture is known when $ n=1$. Assume for the rest of this paragraph that $ \rho$ is odd, i.e., if $ c\in\Gal (\overline{\mathbf{Q}}/\mathbf{Q})$ is complex conjugation, then $ \det(\rho(c))=-1$. When $ n=2$ and the image of $ \rho$ in $ \PGL _2(\mathbf{C})$ is a solvable group, the conjecture is known, and is a deep theorem of Langlands and others (see [Lan80]), which played a crucial roll in Wiles's proof of Fermat's Last Theorem. When $ n=2$ and the image of $ \rho$ in $ \PGL _2(\mathbf{C})$ is not solvable, the only possibility is that the projective image is isomorphic to the alternating group $ A_5$. Because $ A_5$ is the symmetry group of the icosahedron, these representations are called icosahedral. In this case, Joe Buhler's Harvard Ph.D. thesis [Buh78] gave the first example in which $ \rho$ was shown to satisfy Conjecture 9.5.3. There is a book [Fre94], which proves Artin's conjecture for 7 icosahedral representation (none of which are twists of each other). Kevin Buzzard and the author proved the conjecture for 8 more examples [BS02]. Subsequently, Richard Taylor, Kevin Buzzard, Nick Shepherd-Barron, and Mark Dickinson proved the conjecture for an infinite class of icosahedral Galois representations (disjoint from the examples) [BDSBT01]. The general problem for $ n=2$ is in fact now completely solved, due to recent work of Khare and Wintenberger [KW08] that proves Serre's conjecture.

William Stein 2012-09-24